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Remarks on a nonlinear nonlocal operator in Orlicz spaces by Correa, Ernesto; de Pablo, Arturo is a scholarly article available to read on EtoBox.

What is Remarks on a nonlinear nonlocal operator in Orlicz spaces about?

We study integral operators $\mathcal{L}u(x)=\int_{\mathbb{R^N}}\psi(u(x)-u(y))J(x-y)\,dy$ of the type of the fractional $p$-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincar\'e inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel $J$ considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$ in a bounded domain $\Omega$, and boundary condition $u\equiv0$ on $\Omega^c$; both cases $f=f(x)$ and $f=f(u)$ are considred, including the generalized eigenvalue problem $f(u)=\lambda\psi(u)$.

Author

Correa, Ernesto; de Pablo, Arturo

Published

2018

Language

EN